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Neutron--Antineutron Oscillations: Discrete Symmetries and Quark

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Neutron--Antineutron Oscillations: Discrete Symmetries and Quark Neutron–Antineutron Oscillations: Discrete Symmetries and Quark Operators Zurab Berezhiani1, 2 and Arkady Vainshtein3, 4 1Dipartimento di Fisica e Chimica, Universit`adell’Aquila, Via Vetoio, 67100 Coppito, L’Aquila, Italy 2INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi, L’Aquila, Italy 3School of Physics and Astronomy and William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA 4Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA We analyze status of C, P and T discrete symmetries in application to neutron-antineutron transitions breaking conservation of baryon charge B by two units. At the level of free particles all these symmetries are preserved. This includes P reflection in spite of the opposite internal parities usually ascribed to neutron and antineutron. Explanation, which goes back to the 1937 papers by E. Majorana and by G. Racah, is based on a definition of parity satisfying P2 = −1, instead of P2 = 1, and ascribing P = i to both, neutron and antineutron. We apply this to C, P and T classification of six-quark operators with |∆B| = 2. It allows to specify operators contributing to neutron-antineutron oscillations. Remaining operators contribute to other |∆B| = 2 processes and, in particular, to nuclei instability. We also show that presence of external magnetic field does not induce any new operator mixing the neutron and antineutron provided that rotational invariance is not broken. 1. A phenomenon of neutron-antineutron oscillation was 2. Let us start with the Dirac Lagrangian suggested by Kuzmin [1] in 1970, and the first theoretical µ model – by Mohapatra and Marshak in 1980 [2]. It is now D = i nγ ∂µn m nn (1) L − under active discussion (for a review, see [3]). A discov- ery of this oscillations would be a clear evidence of baryon with the four-component spinor nα , (α = 1, ..., 4) and charge nonconservation, ∆ = 2. In this note we dis- the mass parameter m which is real and positive. The cuss the issue of C, P and| TB|symmetries in the ∆ =2 Lagrangian gives the Lorentz-invariant description of free transitions, applying this to analysis of six-quark| B| opera- neutron and antineutron states and preserves the baryon tors. We also analyze effects of external magnetic field charge, = 1 for n and = 1 for n. Its conservation B B − and show that it does not add any new ∆ = 2 operator is associated with the continuous U(1)B symmetry if the rotational invariance is not broken.| B| n eiαn, n e−iα n (2) Essentially the same issues were addressed in our pre- → → vious note [4]. There we emphasize the point that parity of Lagrangian (1). Correspondingly, at each spatial P, defined in such a way that P2 =1, is broken, as well as momentum there are four degenerate states, the spin CP, in the neutron-antineutron transition. This is an im- doublet of the neutron states with the baryon charge mediate consequence of the opposite parities of neutron = 1, and the spin doublet of the antineutron states and antineutron when P2 =1. Indeed, we deal then with withB = 1, i.e., two spin doublets which differ by the mixing of the states with different parities. Although we baryonB charge− . also noted that in the absence of interaction it does not Note that anotherB bilinear mass term, automatically imply an existence of CP breaking physics we did not present a detailed analysis of the problem. We im nγ5n , (3) have corrected this at the INT workshop in September − arXiv:1809.00997v1 [hep-ph] 30 Aug 2018 2015, defining P such that P 2 = 1. z z − consistent with the baryone charge conservation, can be Following our note [4] the issue of parity definition in rotated away by chiral U(1) transformation n eiηγ5 n . the ∆ = 2 transitions was addressed in a number of How the baryon number non-conservation→ shows up | B| related publications [5–7]. Unfortunately, together with at the level of free one-particle states? In Lagrangian correct statements some of these analyses are clearly er- description it could be only modification of the bilinear roneous. For instance, McKeen and Nelson in their inter- mass terms. Generically, there are four such Lorentz in- esting paper [6] about CP violation due to baryon oscil- variant bilinear terms: lations wrongly insisted that one can keep P2 =1 for the T T T T parity definition. It shows that the subject deserves a fur- n Cn, n Cγ5n , nC n , nCγ5n¯ . (4) ther discussion. Actually, the issue of parity definition for fermions was resolved long ago. Below we present more Here C = iγ2γ0 is the charge conjugation matrix in the details of parity definition story which has been started in Dirac (standard) representation of gamma matrices. It 1937 by Ettore Majorana in his famous paper [8] where has the same form in the Weyl (chiral) representation. he introduced a notion of Majorana fermions. In the In the Majorana representation C = γ0. same journal issue the parity definition was discussed in Using the chiral basis we show in− the part 4 that all more details by Giulio Racah [9]. these modifications (4) are reduced by field redefinitions 2 to just one possibility for the baryon charge breaking by eigenvalues of P are 1 and opposite for fermion and two units, antifermion states. ± Different parities of neutron and antineutron imply 1 T T P ∆ 6B = ǫ n Cn + nCn , (5) that their mixing breaks parity, and, indeed, the sub- L −2 stitution (11) changes ∆ 6B to ( ∆ 6B) . Together with L − L where ǫ is a real positive parameter. The possibility C invariance it implies then that ∆ 6B is also CP odd. L of such redefinitions is based on U(2) symmetry of the However, this CP oddness does not translate immedi- µ CP kinetic term i nγ ∂µn. Four-parametric U(2) transfor- ately into observable breaking effects. To get them mations allow to exclude the term (3) and to reduce four one needs an interference of amplitudes and this is pro- terms (4) to just one structure (5). vided only when interaction is present. It shows a subtlety in the definition of parity transfor- P 3. What is the status of discrete C, P and T symmetries mation , see textbook discussions, e.g., in Refs. [11, 12]. under the baryon charge breaking modification (5)? Let Let us remind it. us first consider the charge conjugation C, which can be When baryon charge is conserved there is no transition P viewed as a plain exchange symmetry between n and nc between sectors with different , and one can combine B P fields, with a baryonic U(1)B phase rotation (2) and define α, c T iBα iα 0 c −iα 0 c C : n n = C n . (6) Pα = P e : n e γ n, n e γ n . (12) ←→ → →− C2 P2 2iBα This is a sort of discrete Z2 symmetry, = 1. The most Of course, then α = e = 1 but the phase is unob- simple it looks in the Majorana representation where servable when is conserved.6 When baryonB charge is not conserved the only rem- nc = n∗ . (7) nant of baryonic U(1)B rotations is Z2 symmetry asso- ciated with changing sign of the fermion field, n n. It is straithforward to verify that both Lagrangians → − above, (1) and (5), are C invariant. Indeed, they could This symmetry is protected: unphysical 2π space rota- be rewritten in the form tion changes the sign of the fermion field. It means that besides the original P 2 = 1 we can consider a different i µ c µ c m c c P P2 = nγ ∂ n + n γ ∂ n nn + n n , parity definition z , such that z = 1. LD 2 µ µ − 2 − (8) Thus, choosing α = π/2 in Eq.(12), we come to a new ǫ c c P ∆ 6B = n n + nn , parity z, L −2 C P = P eiBπ/2 : n iγ0n, nc iγ0nc (13) which makes their invariance explicit. z → → The Lagrangians are diagonalized in terms of Majo- with P2 = 1. Now P parities of n and nc states are rana fields n1,2 , z z the same and− equal to i, so their mixing does not break c n n Pz parity. It means that all discrete symmetries, C, Pz n1,2 = ± , (9) T √2 and are preserved by the baryon breaking term ∆ 6B . Couple of related comments. First, one can chooseL which are even and odd under the charge conjugation C, c α = π/2 and have parities of fermion and antifermion n = n1 2. Namely, − 1,2 ± , both equal to ( i) instead of i. The absolute sign has no physical meaning− – it could be changed by a 2π 1 µ D = nkγ ∂µnk m nknk , space rotation – but relative parity between two dif- L 2 − kX=1,2 (10) ferent fermions does make sense. Second, it is amus- c 1 ing that the same Pz parity for n and n equal to i ∆ 6B = ǫ n1 n1 n2 n2 . L −2 − is still consistent with the notion of opposite parities of fermion and antifermion, having in mind that that for It demonstrates that the baryon charge breaking leads to the complex value of parity we should compare Pz(n) splitting into two Majorana spin doublets. The C-even c ∗ with [Pz(n )] . Also for a fermion-antifermion pair the n1 field gets the mass M1 = m + ǫ while the mass of the c product Pz(n)Pz(n ) = 1. One more comment is to C-odd n2 is M2 = m ǫ.
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